Making Maps With R

Making Maps with R

New Packages

• ggmap
• mapdata

Intro

For a long time, R has had a relatively simple mechanism, via the maps package, for making simple outlines of maps and plotting lat-long points and paths on them.

More recently, with the advent of packages like sp, rgdal, and rgeos, R has been acquiring much of the functionality of traditional GIS packages (like ArcGIS, etc). This is an exciting development, but not always easily accessible for the beginner, as it requires installation of specialized external libraries (that may, on some platforms, not be straightforward) and considerable familiarity with GIS concepts.

More recently, a third approach to convenient mapping, using ggmap has been developed that allows the tiling of detailed base maps from Google Earth or Open Street Maps, upon which spatial data may be plotted. Today, we are going to focus on mapping using base maps from R’s tried and true maps package and also using the ggmap package. We won’t cover the more advanced GIS-related topics nor using rgdal, or sp to plot maps with different projections, etc. Nor will cover the somewhat more simplified approach to projections using the mapproj package.

As in our previous explorations in this course, when it comes to plotting, we are going to completely skip over R’s base graphics system and head directly to Hadley Wickham’s ggplot2 package. Hadley has included a few functions that make it relatively easy to interact with the data in R’s maps package, and of course, once a map layer is laid down, you have all the power of ggplot at your fingertips to overlay whatever you may want to over the map. ggmap is a package that goes out to different map servers and grabs base maps to plot things on, then it sets up the coordinate system and writes it out as the base layer for further ggplotting. It is pretty sweet, but does not support different projections.

Today’s Goals

1. Introduce readers to the map outlines available in the maps package
   • Show how to convert those data into data frames that ggplot2 can deal with
   • Discuss some ggplot2 related issues about plotting things.
2. Use ggmap to make some pretty decent looking maps

I feel that the above twp topics should cover a large part of what people will need for making useful maps of field sites, or sampling locations, or fishing track lines, etc.

For today we will be skipping how to read in traditional GIS “shapefiles” so as to minimize the number of packages that need installation, but keep in mind that it isn’t too hard to do that in R, too.

Plotting maps-package maps with ggplot

The main players:

• The maps package contains a lot of outlines of continents, countries, states, and counties that have been with R for a long time.
  
• The mapdata package contains a few more, higher-resolution outlines.
• The maps package comes with a plotting function, but, we will opt to use ggplot2 to plot the maps in the maps package.
  
• Recall that ggplot2 operates on data frames. Therefore we need some way to translate the maps data into a data frame format that ggplot can use.

Maps in the maps package

• Package maps provides lots of different map outlines and points for cities, etc.
  
• Some examples: usa, nz, state, world, etc.

Making data frames from map outlines

• ggplot2 provides the map_data() function.
  • Think of it as a function that turns a series of points along an outline into a data frame of those points.
  • Syntax: map_data("name") where “name” is a quoted string of the name of a map in the maps or mapdata package
• Here we get a USA map from maps:

usa <- map_data("usa")

dim(usa)

## [1] 7243    6

head(usa)

##        long      lat group order region subregion
## 1 -101.4078 29.74224     1     1   main      <NA>
## 2 -101.3906 29.74224     1     2   main      <NA>
## 3 -101.3620 29.65056     1     3   main      <NA>
## 4 -101.3505 29.63911     1     4   main      <NA>
## 5 -101.3219 29.63338     1     5   main      <NA>
## 6 -101.3047 29.64484     1     6   main      <NA>

tail(usa)

##           long      lat group order         region subregion
## 7247 -122.6187 48.37482    10  7247 whidbey island      <NA>
## 7248 -122.6359 48.35764    10  7248 whidbey island      <NA>
## 7249 -122.6703 48.31180    10  7249 whidbey island      <NA>
## 7250 -122.7218 48.23732    10  7250 whidbey island      <NA>
## 7251 -122.7104 48.21440    10  7251 whidbey island      <NA>
## 7252 -122.6703 48.17429    10  7252 whidbey island      <NA>

• Here is the high-res world map centered on the Pacific Ocean from mapdata

w2hr <- map_data("world2Hires")

dim(w2hr)

## [1] 2274539       6

head(w2hr)

##       long      lat group order region subregion
## 1 226.6336 58.42416     1     1 Canada      <NA>
## 2 226.6314 58.42336     1     2 Canada      <NA>
## 3 226.6122 58.41196     1     3 Canada      <NA>
## 4 226.5911 58.40027     1     4 Canada      <NA>
## 5 226.5719 58.38864     1     5 Canada      <NA>
## 6 226.5528 58.37724     1     6 Canada      <NA>

tail(w2hr)

##             long      lat group   order      region subregion
## 2276817 125.0258 11.18471  2284 2276817 Philippines     Leyte
## 2276818 125.0172 11.17142  2284 2276818 Philippines     Leyte
## 2276819 125.0114 11.16110  2284 2276819 Philippines     Leyte
## 2276820 125.0100 11.15555  2284 2276820 Philippines     Leyte
## 2276821 125.0111 11.14861  2284 2276821 Philippines     Leyte
## 2276822 125.0155 11.13887  2284 2276822 Philippines     Leyte

The structure of those data frames

These are pretty straightforward:

• long is longitude. Things to the west of the prime meridian are negative.
• lat is latitude.
• order. This just shows in which order ggplot should “connect the dots”
• region and subregion tell what region or subregion a set of points surrounds.
• group. This is very important! ggplot2’s functions can take a group argument which controls (amongst other things) whether adjacent points should be connected by lines. If they are in the same group, then they get connected, but if they are in different groups then they don’t.
  • Essentially, having to points in different groups means that ggplot “lifts the pen” when going between them.

Plot the USA map

• Maps in this format can be plotted with the polygon geom. i.e. using geom_polygon().
• geom_polygon() drawn lines between points and “closes them up” (i.e. draws a line from the last point back to the first point)
• You have to map the group aesthetic to the group column
• Of course, x = long and y = lat are the other aesthetics.

Simple black map

By default, geom_polygon() draws with no line color, but with a black fill:

usa <- map_data("usa") # we already did this, but we can do it again
ggplot() + 
  geom_polygon(data = usa, aes(x=long, y = lat, group = group)) + 
  coord_fixed(1.3)

What is this coord_fixed()?

• This is very important when drawing maps.
• It fixes the relationship between one unit in the \(y\) direction and one unit in the \(x\) direction.
• Then, even if you change the outer dimensions of the plot (i.e. by changing the window size or the size of the pdf file you are saving it to (in ggsave for example)), the aspect ratio remains unchanged.
• In the above case, I decided that if every \(y\) unit was 1.3 times longer than an \(x\) unit, then the plot came out looking good.
  • A different value might be needed closer to the poles.

Mess with line and fill colors

• Here is no fill, with a red line. Remember, fixed value of aesthetics go outside the aes function.

ggplot(usa) + 
  geom_polygon(aes(x = long, y = lat, group = group), 
               fill = NA, color = "red") + 
  coord_fixed(1.3)

• Here is violet fill, with a blue line.

gg1 <- ggplot(usa) + 
  geom_polygon(aes(x = long, y = lat, group = group), 
               fill = "violet", color = "blue") + 
  coord_fixed(1.3)
gg1

Adding points to the map

• Let’s add black and yellow points at our lab and at the NWFSC in Seattle.

labs <- data.frame(
  long = c(-122.064873, -122.306417),
  lat = c(36.951968, 47.644855),
  names = c("SWFSC-FED", "NWFSC"),
  stringsAsFactors = FALSE
  )  

gg1 + 
  geom_point(data = labs, aes(x = long, y = lat), color = "black", size = 5) +
  geom_point(data = labs, aes(x = long, y = lat), color = "yellow", size = 4)

See how important the group aesthetic is

To show how important the group aesthetic is, here we plot the map without using the group aesthetic:

ggplot() + 
  geom_polygon(data = usa, aes(x = long, y = lat), fill = "violet", color = "blue") + 
  geom_point(data = labs, aes(x = long, y = lat), color = "black", size = 5) +
  geom_point(data = labs, aes(x = long, y = lat), color = "yellow", size = 4) +
  coord_fixed(1.3)

State maps

We can also get a data frame of polygons that tell us above state boundaries:

states <- map_data("state")
dim(states)

## [1] 15537     6

head(states)

##        long      lat group order  region subregion
## 1 -87.46201 30.38968     1     1 alabama      <NA>
## 2 -87.48493 30.37249     1     2 alabama      <NA>
## 3 -87.52503 30.37249     1     3 alabama      <NA>
## 4 -87.53076 30.33239     1     4 alabama      <NA>
## 5 -87.57087 30.32665     1     5 alabama      <NA>
## 6 -87.58806 30.32665     1     6 alabama      <NA>

tail(states)

##            long      lat group order  region subregion
## 15594 -106.3295 41.00659    63 15594 wyoming      <NA>
## 15595 -106.8566 41.01232    63 15595 wyoming      <NA>
## 15596 -107.3093 41.01805    63 15596 wyoming      <NA>
## 15597 -107.9223 41.01805    63 15597 wyoming      <NA>
## 15598 -109.0568 40.98940    63 15598 wyoming      <NA>
## 15599 -109.0511 40.99513    63 15599 wyoming      <NA>

Plot all the states, all colored a little differently

This is just like it is above, but we can map fill to region and make sure the the lines of state borders are white.

ggplot(states) + 
  geom_polygon(aes(x = long, y = lat, fill = region, group = group), color = "white") + 
  coord_fixed(1.3) +
  guides(fill = FALSE)  # do this to leave off the color legend

Boom! That is easy.

Plot just a subset of states in the contiguous 48:

• We can use it to grab just CA, OR, and WA:

west_coast <- filter(states, region %in% c("california", "oregon", "washington"))

ggplot(west_coast) + 
  geom_polygon(aes(x = long, y = lat), fill = "palegreen", color = "black") 

Man that is ugly!!

• I am just keeping people on their toes. What have we forgotten here?
  • group
  • coord_fixed()
• Let’s put those back in there:

ggplot(data = west_coast) + 
  geom_polygon(aes(x = long, y = lat, group = group), fill = "palegreen", color = "black") + 
  coord_fixed(1.3)

Phew! That is a little better!

Zoom in on California and look at counties

• Getting the california data is easy:

ca_df <- filter(states, region == "california")

head(ca_df)

##        long      lat group order     region subregion
## 1 -120.0060 42.00927     4   667 california      <NA>
## 2 -120.0060 41.20139     4   668 california      <NA>
## 3 -120.0060 39.70024     4   669 california      <NA>
## 4 -119.9946 39.44241     4   670 california      <NA>
## 5 -120.0060 39.31636     4   671 california      <NA>
## 6 -120.0060 39.16166     4   672 california      <NA>

• Now, let’s also get the county lines there

counties <- map_data("county")
ca_county <- filter(counties, region == "california")

head(ca_county)

##        long      lat group order     region subregion
## 1 -121.4785 37.48290   157  6965 california   alameda
## 2 -121.5129 37.48290   157  6966 california   alameda
## 3 -121.8853 37.48290   157  6967 california   alameda
## 4 -121.8968 37.46571   157  6968 california   alameda
## 5 -121.9254 37.45998   157  6969 california   alameda
## 6 -121.9483 37.47717   157  6970 california   alameda

• Plot the state first but let’s ditch the axes gridlines, and gray background by using the super-wonderful theme_nothing().

ca_base <- ggplot(ca_df, aes(x = long, y = lat, group = group)) + 
  coord_fixed(1.3) + 
  geom_polygon(color = "black", fill = "gray")
ca_base + theme_nothing()

• Now plot the county boundaries in white:

ca_base + theme_nothing() + 
  geom_polygon(data = ca_county, fill = NA, color = "white") +
  geom_polygon(color = "black", fill = NA)  # get the state border back on top

Get some facts about the counties

• The above is pretty cool, but it seems like it would be a lot cooler if we could plot some information about those counties.
  
• Now I can go to wikipedia and grab population and area data for each county.

# scrape population data
county_html <- read_html("https://en.wikipedia.org/wiki/List_of_counties_in_California")
pop_and_area <- county_html %>% 
  html_node(".wikitable.sortable") %>% 
  html_table() %>% 
  select(county = County, population = `Population[6]`, area = `Area[4]`) %>% 
  mutate(county = county %>% 
           str_replace_all(c("City and County of San Francisco" = "San Francisco",
                         " County" = "")) %>% 
           str_to_lower,
         population = population %>% 
           str_replace_all(".*\u2660", "") %>% parse_number,
         area = area %>% 
           str_replace_all(".*\u2660|\n.*", "") %>%  parse_number)

• We now have the numbers that we want, but we need to attach those to every point on polygons of the counties. This is a job for inner_join.

cacopa <- inner_join(ca_county, pop_and_area, by = c("subregion" = "county"))

• And finally, add a column of people_per_mile:

cacopa <- mutate(cacopa, people_per_mile = population / area)

head(cacopa)

##        long      lat group order     region subregion population area
## 1 -121.4785 37.48290   157  6965 california   alameda    1638215  738
## 2 -121.5129 37.48290   157  6966 california   alameda    1638215  738
## 3 -121.8853 37.48290   157  6967 california   alameda    1638215  738
## 4 -121.8968 37.46571   157  6968 california   alameda    1638215  738
## 5 -121.9254 37.45998   157  6969 california   alameda    1638215  738
## 6 -121.9483 37.47717   157  6970 california   alameda    1638215  738
##   people_per_mile
## 1        2219.804
## 2        2219.804
## 3        2219.804
## 4        2219.804
## 5        2219.804
## 6        2219.804

Now plot population density by county

If you were needing a little more elbow room in the great Golden State, this shows you where you can find it:

# prepare to drop the axes and ticks but leave the guides and legends
# We can't just throw down a theme_nothing()!
ditch_the_axes <- theme(
  axis.text = element_blank(),
  axis.line = element_blank(),
  axis.ticks = element_blank(),
  panel.border = element_blank(),
  panel.grid = element_blank(),
  axis.title = element_blank()
  )

elbow_room1 <- ca_base + 
      geom_polygon(data = cacopa, aes(fill = people_per_mile), color = "white") +
      geom_polygon(color = "black", fill = NA) +
      theme_bw() +
      ditch_the_axes

elbow_room1 

Lame!

• The popuation density in San Francisco is so great that it makes it hard to discern differences between other areas.
• This is a job for a scale transformation. Let’s take the log-base-10 of the population density.
• Instead of making a new column which is log10 of the people_per_mile we can just apply the transformation in the gradient using the trans argument

elbow_room1 + scale_fill_gradient(trans = "log10")

Still not great

It’s very common for color to represent a dimension of your data in spatial visualizations. As such, you should think about the color palette carefully. Here, contrasting colors are used to show the gamut of population density in California.

eb2 <- elbow_room1 + 
    scale_fill_gradient(low = "#0C374D", high = "#EFD469", trans = "log10")
eb2

Zooming in?

Note that the scale of these maps from package maps are not great. We can zoom in to the Bay region, and it sort of works scale-wise, but if we wanted to zoom in more, it would be tough.

Let’s try!

eb2 + xlim(-123, -121.0) + ylim(36, 38)

• Welp, that was a big dose of derpatitis. Why?
• Recall that geom_polygon() connects the end point of a group to its starting point.
• xlim and ylim functions in ggplot2 discard all the data that is not within the plot area.
  • Hence there are new starting points and ending points for some groups (or in this case the black-line permiter of California) and those points get connected. Not good.

True zoom.

• If you want to keep all the data the same but just zoom in, you can use the xlim and ylim arguments to coord_cartesian(). Though, to keep the aspect ratio correct we must use coord_fixed() instead of coord_cartesian().
• This chops stuff off but doesn’t discard it from the data set:

eb2 + coord_fixed(xlim = c(-123, -121.0),  
                  ylim = c(36, 38), 
                  ratio = 1.3)

ggmap

The ggmap package is the most exciting R mapping tool in a long time. You might be able to get better looking maps at some resolutions by using shapefiles and rasters from naturalearthdata.com but ggmap will get you 95% of the way there with only 5% of the work!

Three examples

1. Named “sampling” points on the Sisquoc River from the “Sisquoctober Adventure”
2. A GPS track from a short bike ride in Wilder Ranch.
3. Fish sampling locations from the coded wire tag data base.

How ggmap works

• ggmap simplifies the process of downloading base maps from Google or Open Street Maps or Stamen Maps to use in the background of your plots.
• It also sets the axis scales, etc, in a nice way.
• Once you have gotten your maps, you make a call with ggmap() much as you would with ggplot()
• Let’s do by example.

Sisquoctober

• Here is a small data frame of points from the Sisquoc River.

sisquoc <- read_tsv("data/sisquoc-points.txt")
sisquoc

## # A tibble: 8 x 3
##     name       lon      lat
##    <chr>     <dbl>    <dbl>
## 1    a17 -119.7603 34.75474
## 2 a20-24 -119.7563 34.75380
## 3 a25-28 -119.7537 34.75371
## 4 a18,19 -119.7573 34.75409
## 5 a35,36 -119.7467 34.75144
## 6    a31 -119.7478 34.75234
## 7    a38 -119.7447 34.75230
## 8    a43 -119.7437 34.75251

Let’s try using the zoom level. Zoom levels go from 3 (world scale to 20 (house scale)).

# compute the mean lat and lon
ll_means <- summarise_each(sisquoc, "mean", lon, lat) %>% as.numeric(.[1, ])
sq_map2 <- get_map(location = ll_means,  maptype = "satellite", 
                   source = "google", zoom = 15)

## Map from URL : http://maps.googleapis.com/maps/api/staticmap?center=34.753117,-119.751324&zoom=15&size=640x640&scale=2&maptype=satellite&language=en-EN&sensor=false

ggmap(sq_map2) + 
  geom_point(data = sisquoc, color = "red", size = 4) +
  geom_text(data = sisquoc, aes(label = name), 
            angle = 60, hjust = -.25, color = "yellow")

* That is decent. How about if we use the “terrain” type of map:

sq_map3 <- get_map(location = ll_means,  maptype = "terrain", 
                   source = "google", zoom = 15)

## Map from URL : http://maps.googleapis.com/maps/api/staticmap?center=34.753117,-119.751324&zoom=15&size=640x640&scale=2&maptype=terrain&language=en-EN&sensor=false

ggmap(sq_map3) + 
  geom_point(data = sisquoc, color = "red", size = 4) +
  geom_text(data = sisquoc, aes(label = name), 
            angle = 60, hjust = -.25, color = "yellow")

• That is cool, but I would search for a better color for the lettering…

How about a bike ride?

• Here we use some GPS data for a bike ride.
• We can plot the route like this:

bike <- read_csv("data/bike-ride.csv")
head(bike)

## # A tibble: 6 x 4
##         lon      lat elevation                time
##       <dbl>    <dbl>     <dbl>              <time>
## 1 -122.0646 36.95144      15.8 2011-12-08 19:37:56
## 2 -122.0646 36.95191      15.5 2011-12-08 19:37:59
## 3 -122.0645 36.95201      15.4 2011-12-08 19:38:04
## 4 -122.0645 36.95218      15.5 2011-12-08 19:38:07
## 5 -122.0643 36.95224      15.7 2011-12-08 19:38:10
## 6 -122.0642 36.95233      15.8 2011-12-08 19:38:13

bikemap1 <- get_map(location = c(-122.080954, 36.971709), 
                    maptype = "terrain", source = "google", zoom = 14)

## Map from URL : http://maps.googleapis.com/maps/api/staticmap?center=36.971709,-122.080954&zoom=14&size=640x640&scale=2&maptype=terrain&language=en-EN&sensor=false

ggmap(bikemap1) + 
  geom_path(data = bike, aes(color = elevation), size = 3, lineend = "round")

• Note that getting the right zoom and position for the map is sort of trial and error. You can go to google maps to figure out where the center should be (right click and choose “What’s here?” to get the lat-long of any point. )

Fish sampling locations

In this example, we use fishery data from Eric Anderson of NOAA. He whittled down some stuff in the coded wire tag data base to georeferenced marine locations in British Columbia where at least one Chinook salmon was recovered in between 2000 and 2012 inclusive. To see how he did all that you can check out this

Let’s have a look at the data:

bc <- readRDS("data/bc_sites.rds")

# look at some of it:
bc %>% select(state_or_province:sub_location, longitude, latitude)

## # A tibble: 1,113 x 9
##    state_or_province water_type sector region  area location sub_location
##                <chr>      <chr>  <chr>  <chr> <chr>    <chr>        <chr>
## 1                  2          M      S     22  016   THOR IS           01
## 2                  2          M      N     26  012   MITC BY           18
## 3                  2          M      S     22  015   HARW IS           02
## 4                  2          M      N     26  006   HOPK PT           01
## 5                  2          M      S     23  017   TENT IS           06
## 6                  2          M      S     28  23A   NAHM BY           02
## 7                  2          M      N     26  006   GIL IS            06
## 8                  2          M      S     27  024   CLEL IS           06
## 9                  2          M      S     27  23B   SAND IS           04
## 10                 2          M      N     26  012   DUVA IS           16
## # ... with 1,103 more rows, and 2 more variables: longitude <dbl>,
## #   latitude <dbl>

So, we have 1,113 points to play with.

What do we hope to learn?

• These locations in BC are hierarchically structured. I am basically interested in how close together sites in the same “region” or “area” or “sector” are, and pondering whether it is OK to aggregate fish recoveries at a certain level for the purposes of getting a better overall estimate of the proportion of fish from different hatcheries in these areas.
• So, pretty simple stuff. I just want to plot these points on a map, and paint them a different color according to their sector, region, area, etc.
• Let’s just enumerate things first:

bc %>% group_by(sector, region, area) %>% tally()

## Source: local data frame [42 x 4]
## Groups: sector, region [?]
## 
##    sector region  area     n
##     <chr>  <chr> <chr> <int>
## 1             48  008      1
## 2             48  028      1
## 3             48  311      1
## 4       N     25  001     33
## 5       N     25  003     15
## 6       N     25  004     44
## 7       N     25  02E      2
## 8       N     25  02W     34
## 9       N     26  006     28
## 10      N     26  007     23
## # ... with 32 more rows

• That looks good. It appears like we could probably color code over the whole area down to region, and then down to area within subregions.

Making a map.

• Let’s try to use make_bbox(). This makes a bounding box rather than relying on the zoom feature. It’s an experimental feature that would be cool if it worked reliably, but unfortunately, it’s known to fail.

# compute the bounding box
bc_bbox <- make_bbox(lat = latitude, lon = longitude, data = bc)
bc_bbox

##       left     bottom      right        top 
## -133.63297   47.92497 -122.33652   55.80833

# grab the maps from google
bc_big <- get_map(location = bc_bbox, source = "google", maptype = "terrain")

## Warning: bounding box given to google - spatial extent only approximate.

## converting bounding box to center/zoom specification. (experimental)

## Map from URL : http://maps.googleapis.com/maps/api/staticmap?center=51.86665,-127.98475&zoom=6&size=640x640&scale=2&maptype=terrain&language=en-EN&sensor=false

# plot the points and color them by sector
ggmap(bc_big) + 
  geom_point(data = bc, 
             mapping = aes(x = longitude, y = latitude, color = sector))

• Cool! That was about as easy as could be. North is in the north, south is in the south, and the three reddish points are clearly aberrant ones at the mouths of rivers.

Coloring it by region

ggmap(bc_big) + 
  geom_point(data = bc, mapping = aes(x = longitude, y = latitude, color = region))

Zooming in on each region and coloring by area

• It is time to really put this thing through its paces. (Keeping in mind that make_bbox() might fail…)
• I want to make series of maps. One for each region, in which the the areas in that region are colored differently.
• How? Let’s make a function: you pass it the region and it makes the plot.

region_plot <- function(my_region) {
  region_data <- bc %>% filter(region == my_region)
  bbox <- make_bbox(lon = longitude, lat = latitude, data = region_data)
  my_map <- get_map(location = bbox, source = "google", maptype = "terrain")
  # now we want to count up how many areas there are
  num_areas <- region_data %>% summarise(n_distinct(area)) %>% as.numeric
  num_points <- nrow(region_data)
  
  the_map <- ggmap(my_map) +
    geom_point(data = region_data, 
               mapping = aes(x = longitude, y = latitude), 
               size = 4, color = "black") +
    geom_point(data = region_data, 
               mapping = aes(x = longitude, 
                             y = latitude, color = area), size = 3) +
    ggtitle(sprintf("BC Region: %s with %d locations in %d area(s)",
                    my_region, num_points, num_areas))
  
  ggsave(sprintf("bc_region%s.pdf", my_region), the_map, width = 9, height = 9)
}

So, with that function we just need to cycle over the regions and make all those plots. Note that I am saving them to PDFs because it is no fun to make a web page with all of those in there.

# invisible suppresses lapply's list output, which we're not interested in
bc$region %>% 
  unique %>% 
  lapply(region_plot) %>%
  invisible

Further Reading

ggmap cheat sheet ggmap documentation Spatial Plots